Some Open Problems Arising from my Recent Finite Field Research

Some Open Problems Arising from my Recent Finite Field Research Gary L. Mullen Penn State University mullen@math.psu.edu July 13, 2015 Some Open Problems Arising from myrecent Finite Field Research July 13, 2015 1 / 37

Let q be a prime power Let F q denote the finite field with q elements Some Open Problems Arising from myrecent Finite Field Research July 13, 2015 2 / 37

E-perfect codes F. Castro, H. Janwa, M, I. Rubio, Bull. ICA (2016) Theorem (Hamming bound) Let C be a t-error-correcting code of length n over F q. Then [ ( ) ( ) ( ] n n n C 1 + (q 1) + (q 1) 2 + + )(q 1) t q n. 1 2 t A code C is perfect if the code s parameters yield an equality in the Hamming bound. The parameters of all perfect codes are known, and can be listed as follows: Some Open Problems Arising from myrecent Finite Field Research July 13, 2015 3 / 37

The trivial perfect codes are 1 The zero vector (0,..., 0) of length n, 2 The entire vector space F n q 3 The binary repetition code of odd length n. The non-trivial perfect codes must have the parameters ( n, M = q k, 3 ) of the Hamming codes and the Golay codes (unique up to equivalence) whose parameters can be listed as follows: 1 The Hamming code positive integer; 2 The [11, 6, 5] Golay code over F 3 ; 3 The [23, 12, 7] Golay code over F 2. [ q m 1 q 1, n m, 3 ] over F q, where m 2 is a Some Open Problems Arising from myrecent Finite Field Research July 13, 2015 4 / 37

Let C be a t-error-correcting code of length n over F q. Then, C [ 1 + ( ) n (q 1) + 1 ( ) n (q 1) 2 + + 2 ( ] n )(q 1) t q n. t A t-error correcting code C with parameters (n, M, d), t = d 1 2, is e-perfect if in the Hamming bound, equality is achieved when, on the right hand side, q n is replaced by q e. An n-perfect code is a perfect code. Some Open Problems Arising from myrecent Finite Field Research July 13, 2015 5 / 37

Conjecture Let C be an (n, M, d) t-error correcting non-trivial e-perfect code over F q. Then C must have one of the following sets of parameters: ( ) q 1 m 1 q 1, qe m, 3, with q a prime power and m < e n, where m 2; 2 (11, 3 e 5, 5), with q = 3 and 5 < e 11; 3 (23, 2 e 11, 7), with q = 2 and 11 < e 23; 4 (90, 2 e 12, 5), with q = 2 and 12 < e 89. Problem Prove this conjecture. We can construct e-perfect codes with each of the parameters listed above, except for the case when n = 90 and e = 89. As was the case for perfect codes, there could be many e-perfect codes with a given set of parameters. Some Open Problems Arising from myrecent Finite Field Research July 13, 2015 6 / 37

R-closed subsets of Z p S. Huczynska, M, J. Yucas, JCT, A (2009) Let G be a finite abelian group with G = g Let S be a subset of G with S = s. Definition Let 0 r s 2. A set S is r-closed if, among the s 2 ordered pairs (a, b) with a, b S, there are exactly r pairs such that a + b S. The r-value of the r-closed set S is denoted by r(s). Some Open Problems Arising from myrecent Finite Field Research July 13, 2015 7 / 37

If S is a subgroup of G then S is s 2 -closed If S is a sum-free set then S is 0-closed. For a given G, what (if anything) can be said about the spectrum of r-values of the subsets of G? Motivated by the classical Cauchy-Davenport Theorem, we are particularly interested in the case when G = Z p under addition modulo the prime p. Some Open Problems Arising from myrecent Finite Field Research July 13, 2015 8 / 37

For G = Z p we characterize the maximal and minimal possible r-values. We make a conjecture (verified computationally for all primes p 23) about the complete spectrum of r-values for any subset cardinality in Z p and prove that, for any p, all conjectured r-values in the spectrum are attained when the subset cardinality is suitably small (s < 2p+2 7 ). Some Open Problems Arising from myrecent Finite Field Research July 13, 2015 9 / 37

Theorem Let G be a finite abelian group of order g. Let s be a positive integer with 0 s g, and let S be a subset of G of size s. Let T be the complement of S in G. Then r(s) + r(t ) = g 2 3gs + 3s 2. Some Open Problems Arising from myrecent Finite Field Research July 13, 2015 10 / 37

Theorem (Cauchy-Davenport) If A and B are non-empty subsets of Z p then A + B min(p, A + B 1). Definition For p be a prime, define Proposition k[p] = p + 1 3 = p 1 3, p 1 mod 3 p 3, p 0 mod 3, p 1 mod 3 p+1 3 Let p be a prime. If S Z p is 0-closed then S k[p]. Some Open Problems Arising from myrecent Finite Field Research July 13, 2015 11 / 37

Definition Let p be an odd prime. For 0 s p, define f s and g s as follows: 0 s k[p] f s = (3s p) 2 1 4 s > k[p] and s even (3s p) 2 4 s > k[p] and s odd 3s 2 4 s p k[p] and s even g s = 3s 2 +1 4 s p k[p] and s odd p 2 3sp + 3s 2 s > p k[p] Note that f s + g p s = p 2 3sp + 3s 2. Some Open Problems Arising from myrecent Finite Field Research July 13, 2015 12 / 37

Proposition Let p > 11. For 1 s 3 and p 3 s p, the r-values for subsets of Z p of size s are precisely the integers in the interval [f s, g s ] with the following exceptions: s f s g s exceptions 1 0 1 2 0 3 2 3 0 7 4 p p 2 p 2 p 1 p 2 3p + 2 p 2 3p + 3 p 2 p 2 6p + 9 p 2 6p + 12 p 2 6p + 10 p 3 p 2 9p + 20 p 2 9p + 27 p 2 9p + 23 Some Open Problems Arising from myrecent Finite Field Research July 13, 2015 13 / 37

Definition If 4 s p 4, define V (s) by 0 if s k[p] V (s) = p s 3 4 if s p+1 3s p 1 4 otherwise 2. Some Open Problems Arising from myrecent Finite Field Research July 13, 2015 14 / 37

Conjecture For p > 11 and 4 s p 4, there are V (s) exceptional values at the low end of the interval [f s, g s ] and V (p s) exceptional values at the high end of the interval [f s, g s ]. All other values in the interval can be obtained as r-values. The exceptions are given by: f s + 3i + 1 for 0 i < V (s) if s p mod 2 f s + 3i + 2 for 0 i < V (s) if s p mod 2 g s 3i 1 for 0 i < V (p s) if s is even g s 3i 2 for 0 i < V (p s) if s is odd Verified computationally for all primes p 23 and all corresponding s (4 s p 4). Problem Prove the conjecture Some Open Problems Arising from myrecent Finite Field Research July 13, 2015 15 / 37

All conjectured r-values in the spectrum are attained when the subset cardinality is suitably small (s < 2p+2 7 ). Some Open Problems Arising from myrecent Finite Field Research July 13, 2015 16 / 37

Subfield Value Sets W.-S. Chou, J. Gomez-Calderon, M, D. Panario, D. Thomson, Funct. Approx. Comment. Math. (2013) Let F q d be a subfield of F q e so d e For f F q e[x], subfield value set V f (q e ; q d ) = {f(c) F q d c F q e} Theorem V x n(q e ; q d ) = (n(qd 1), q e 1) (n, q e 1) + 1 Some Open Problems Arising from myrecent Finite Field Research July 13, 2015 17 / 37

Dickson poly. deg. n, parameter a F q D n (x, 0) = x n Theorem D n (x, a) = n/2 i=0 n n i Chou, Gomez-Calderon, M, JNT, (1988) α usually 0. ( n i i ) ( a) i x n 2i V Dn(x,a) = q 1 2(n, q 1) + q + 1 2(n, q + 1) + α Some Open Problems Arising from myrecent Finite Field Research July 13, 2015 18 / 37

Theorem q odd and a F q e with an F q d, η q e(a) = 1 and η q d(a n ) = 1, Problem V Dn(x,a)(q e ; q d ) = (qe 1, n(q d 1)) + (q e 1, n(q d + 1)) 2(q e 1, n) + (qe + 1, n(q d 1)) + (q e + 1, n(q d + 1)) 2(q e + 1, n) 3 + ( 1)n+1 2 Find subfield value set V Dn(x,a)(q e ; q d ) when a F q e and an F q d Some Open Problems Arising from myrecent Finite Field Research July 13, 2015 19 / 37

In order to have D n (c, a) = y n + an y n F q d we need (y n + an y n )qd = y n + an y n. If a n F q d (y n(qd 1) 1)(y n(qd +1) a n ) = 0. Some Open Problems Arising from myrecent Finite Field Research July 13, 2015 20 / 37

Hypercubes of class r J. Ethier, M, D. Panario, B. Stevens, D. Thomson, JCT, A (2011) Definition Let d, n, r, t be integers, with d > 0, n > 0, r > 0 and 0 t d r. A (d, n, r, t)-hypercube of dimension d, order n, class r and type t is an n n (d times) array on n r distinct symbols such that in every t-subarray (that is, fix t coordinates of the array and allow the remaining d t coordinates to vary) each of the n r distinct symbols appears exactly n d t r times. If d 2r, two such hypercubes are orthogonal if when superimposed, each of the n 2r possible distinct pairs occurs exactly n d 2r times. A set H of such hypercubes is mutually orthogonal if any two distinct hypercubes in H are orthogonal. A (2, n, 1, 1) hypercube is a latin square order n. If r = 1 we have latin hypercubes. Some Open Problems Arising from myrecent Finite Field Research July 13, 2015 21 / 37

0 1 2 4 5 3 8 6 7 3 4 5 7 8 6 2 0 1 6 7 8 1 2 0 5 3 4 A hypercube of dimension 3, order 3, class 2, and type 1. Some Open Problems Arising from myrecent Finite Field Research July 13, 2015 22 / 37

Theorem The maximum number of mutually orthogonal hypercubes of dimension d, order n, type t, and class r is bounded above by ( ( ) ( ) ( ) 1 d d d n r n d 1 (n 1) (n 1) 2 )(n 1) t. 1 1 2 t Corollary There are at most n 1 mutually orthogonal Latin squares of order n. Theorem Let q be a prime power. The number of (2r, q, r, r)-hypercubes is at least the number of linear MDS codes over F q of length 2r and dimension r. Some Open Problems Arising from myrecent Finite Field Research July 13, 2015 23 / 37

Theorem There are at most (n 1) r, (2r, n, r, r) mutually orthogonal hypercubes. Theorem Let n be a prime power. For any integer r < n, there is a set of n 1 mutually orthogonal (2r, n, r, r)-hypercubes. Theorem Let n = 2 2k, k N. Then there is a complete set of (n 1) 2 mutually orthogonal hypercubes of dimension 4, order n, and class 2. D. Droz: If r = 2 and n is odd, there is complete set. Some Open Problems Arising from myrecent Finite Field Research July 13, 2015 24 / 37

Hypercube problems 1 Construct a complete set of mutually orthogonal (4, n, 2, 2)-hypercubes when n = 2 2k+1. D. Droz: If r = 2, n = 2 2k+1 there are (n 1)(n 2) MOHC. Are there (n 1) 2 MOHC? 2 Is the (n 1) r bound tight when r > 2? If so, construct a complete set of mutually orthogonal (2r, n, r, r)-hypercubes of class r > 2. If not, determine a tight upper bound and construct such a complete set. D. Droz: If r 1 and n 1 (mod r), there is complete set. D. Droz: If n = p rk there is a complete set. 3 Find constructions (other than the standard Kronecker product constructions) for sets of mutually orthogonal hypercubes when n is not a prime power. Such constructions will require a new method not based on finite fields. 4 What can be said when d > 2r? Some Open Problems Arising from myrecent Finite Field Research July 13, 2015 25 / 37

k-normal elements S. Huczynska, M, D. Panario, D. Thomson, FFA (2013) Let q be a prime power and n N. An element α F q n yields a normal basis for F q n over F q if B = {α, α q,..., α qn 1 } is a basis for F q n over F q ; such an α is a normal element of F q n over F q. Some Open Problems Arising from myrecent Finite Field Research July 13, 2015 26 / 37

Theorem For α F q n, {α, α q,..., α qn 1 } is a normal basis for F q n over F q if and only if the polynomials x n 1 and αx n 1 + α q x n 2 + + α qn 1 in F q n[x] are relatively prime. Motivated by this, we make the Definition Let α F q n. Denote by g α (x) the polynomial n 1 i=0 αqi x n 1 i F q n[x]. If gcd(x n 1, g α (x)) over F q n has degree k (where 0 k n 1), then α is a k-normal element of F q n over F q. A normal element of F q n over F q is 0-normal. Some Open Problems Arising from myrecent Finite Field Research July 13, 2015 27 / 37

Definition Let f F q [x] be monic, the Euler Phi function for polynomials is given by Φ q (f) = (F q [x]/ff q [x]). Theorem The number of k-normal elements of F q n h x n 1, deg(h)=n k over F q is given by Φ q (h), (1) where divisors are monic and polynomial division is over F q. Some Open Problems Arising from myrecent Finite Field Research July 13, 2015 28 / 37

An important extension of the Normal Basis Theorem is the Primitive Normal Basis Theorem which establishes that, for all pairs (q, n), a normal basis {α, α q,..., α qn 1 } for F q n over F q exists with α a primitive element of F q n. We ask whether an analogous claim can be made about k-normal elements for certain non-zero values of k? In particular, when k = 1, does there always exist a primitive 1-normal element of F q n over F q? Some Open Problems Arising from myrecent Finite Field Research July 13, 2015 29 / 37

Theorem Let q = p e be a prime power and n N with p n. Assume that n 6 if q 11, and that n 3 if 3 q 9. Then there exists a primitive 1-normal element of F q n over F q. Problem Obtain a complete existence result for primitive 1-normal elements of F q n over F q (with or without a computer). We conjecture that such elements always exist. Some Open Problems Arising from myrecent Finite Field Research July 13, 2015 30 / 37

Problem For which values of q, n and k can explicit formulas be obtained for the number of k-normal primitive elements of F q n over F q? Problem Determine the pairs (n, k) such that there exist primitive k-normal elements of F q n over F q. Some Open Problems Arising from myrecent Finite Field Research July 13, 2015 31 / 37

Conjecture (L. Anderson/M) Let p 5 be a prime and let m 3. Let a be 1 or 2 and let k be 0 or 1. Then there is an element α F p m of order pm 1 a which is k-normal. The a = 1, k = 0 case gives the Prim. Nor. Basis Thm. Problem Determine the existence of high-order k-normal elements α F q n over F q. Some Open Problems Arising from myrecent Finite Field Research July 13, 2015 32 / 37

Dickson Polynomials Dickson poly. deg. n, parameter a F q D n (x, 0) = x n D n (x, a) = n/2 i=0 n n i ( n i i ) ( a) i x n 2i Some Open Problems Arising from myrecent Finite Field Research July 13, 2015 33 / 37

Theorem Nöbauer (1968) For a 0, D n (x, a) PP on F q iff (n, q 2 1) = 1. Theorem Chou, Gomez-Calderon, M, JNT, (1988) V Dn(x,a) = q 1 2(n, q 1) + q + 1 2(n, q + 1) + α α usually 0 Some Open Problems Arising from myrecent Finite Field Research July 13, 2015 34 / 37

Reverse Dickson Polynomials Fix x F q and let a be the variable in D n (x, a) Some basic PP results on RDPs in Hou, Sellers, M, Yucas, FFA, 2009 f : F q F q is almost perfect nonlinear (APN) if for each a Fq b F q the eq. f(x + a) f(x) = b has at most two solutions in F q and Theorem For p odd, x n APN on F p 2e implies x n APN on F p e implies D n (1, x) PP on F p e Some Open Problems Arising from myrecent Finite Field Research July 13, 2015 35 / 37

Conjecture Let p > 3 be a prime and let 1 n p 2 1. Then D n (1, x) is a PP on F p if and only if 2, 2p, 3, 3p, p + 1, p + 2, 2p + 1 if p 1 (mod 12), 2, 2p, 3, 3p, p + 1 if p 5 (mod 12), n = 2, 2p, 3, 3p, p + 2, 2p + 1 if p 7 (mod 12), 2, 2p, 3, 3p if p 11 (mod 12). Problem Complete the PP classification for RDPs over F p. Problem What happens over F q when q is a prime power? Problem Determine value set for RDPs over F p Some Open Problems Arising from myrecent Finite Field Research July 13, 2015 36 / 37

THANK YOU!!! Some Open Problems Arising from myrecent Finite Field Research July 13, 2015 37 / 37